Weakened Ramsey Theory

Weakened Ramsey Theory provides readers with an overview of weakened generalizations of various parameters studied in Ramsey theory. Rather than determining how large a structure must be to guarantee the existence of monochromatic substructures in a t-coloring, weakened Ramsey theory considers how large a structure must be to guarantee the existence of a substructure that uses at most a given number of colors. Weakened generalizations of Ramsey numbers, Gallai-Ramsey numbers, Schur numbers, Rado numbers, rainbow Schur numbers, and others have all been studied in the literature, and this book offers a thorough overview of the known results, with complete proofs.

This book should prove a valuable resource to advanced undergraduates, graduate students, and researchers working in Ramsey theory. Only a basic background in graph theory, algebra, and number theory is assumed, although a concise review of important definitions and background results is included.

Features

• Self-contained treatment of the subject, requiring only a minimal background in graph theory, algebra, and number theory
• Complete proofs of major results in weakened Ramsey theory
• Numerous color figures to assist the reader
• Comprehensive list of references
• Tables of known weakened parameters in Ramsey theory, providing a useful reference for researchers.

Mark Rowland Budden completed his Ph.D. in Mathematics in 2003 at the University of Missouri-Columbia. He was a faculty member at Armstrong Atlantic State University (now Georgia Southern University’s Armstrong campus) until 2010, when he joined the faculty at Western Carolina University, where he is now a professor. He has published 47 research articles and is the author of the book Star-Critical Ramsey Numbers for Graphs. Most of his research is in Ramsey theory or other related topics in combinatorial number theory, and the majority of his publications have included student coauthors.